Problem: Rewrite as a simplified fraction. ${1.\overline{65} = {?}}$
Explanation: Let $x$ equal the decimal. Set up two equations such that the digits after the decimal point are identical. ${\begin{align*}100x &= 165.6565...\\ 1x &= 1.6565...\end{align*}}$ Subtracting the two equations, we have: ${99x = 164}$ ${x = \dfrac{164}{99}} \text{or} {1 \dfrac{65}{99}}$